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Chapter 10: Problem 33

Add or subtract. \(-2 \sqrt[4]{x^{7}}+3 \sqrt[4]{16 x^{7}}\)

Short Answer

Expert verified
The simplified expression is \( 4 \sqrt[4]{x^7} \).

Step by step solution

01

Simplify the Radicals

First, simplify the expression inside the radicals. The term \( \sqrt[4]{x^7} \) cannot be simplified further because \( 7 \) is not a multiple of \( 4 \). For the term \( \sqrt[4]{16x^7} \), we can simplify: \( 16 \) is \( 2^4 \), so \( \sqrt[4]{16} = 2 \). Thus, \( \sqrt[4]{16x^7} = 2 \sqrt[4]{x^7} \).
02

Substitute the Simplified Terms

Now substitute the simplified radicals back into the expression. The original expression \( -2 \sqrt[4]{x^7} + 3 \sqrt[4]{16x^{7}} \) becomes \( -2 \sqrt[4]{x^7} + 3 \times 2 \sqrt[4]{x^7} \), which is \( -2 \sqrt[4]{x^7} + 6 \sqrt[4]{x^7} \).
03

Perform Addition

Combine the like terms. Since both terms involve \( \sqrt[4]{x^7} \), simply add the coefficients: \( -2 + 6 = 4 \). Thus, the simplified expression is \( 4 \sqrt[4]{x^7} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Addition of Radicals
Adding radicals might seem complex at first, but it's not too different from adding other types of algebraic terms. When we talk about adding radicals, we mean specifically the process of combining radical expressions that have the same index and radicand. The radicand is the expression inside the radical, and the index is the type of root you are taking.
  • To add radicals, first ensure that both radicals are like radicals. "Like radicals" have the same radicand and index, just like like-terms in algebra.
  • In the problem, you have \(-2 \sqrt[4]{x^7} + 3 \sqrt[4]{16 x^7}\), both terms have the same index and the same radicand involving \(x^7\), but \(\sqrt[4]{16}\) simplifies further.
  • After simplifying, simply add the coefficients in front of the radicals, as demonstrated in the solution.
Simplifying Radicals
By simplifying radicals, you transform a complex radical into its simplest form. This often involves factoring the radicand and simplifying any perfect powers. Let's break it down:
  • The process starts by examining the radicand, looking for factors that are powers of the index. In our example, \(16 \) was used, which simplifies to \(2^4\). Hence, \(\sqrt[4]{16} = 2\).
  • It's essential to simplify any numerical parts first, before proceeding to variables or more complex expressions.
  • The simplified form enhances ease in further operations such as addition or subtraction of radicals.
Remember, if a radical cannot be simplified, you can leave it as it is, just like \(\sqrt[4]{x^7}\) in this problem.
Combining Like Terms
Combining like terms is one of the easiest ways to simplify an expression. Like terms are terms whose variables and exponents (if any) are the same.
  • In radical expressions, once you have like radicals (same radicand and index), add or subtract their coefficients just like any other like terms.
  • In the solution you performed \(-2 + 6\) for the terms \(-2 \sqrt[4]{x^7}\ and 6 \sqrt[4]{x^7}\).
  • This results in \(4 \sqrt[4]{x^7}\), combining the coefficients gives a single term that is much simpler to work with.
Combining like terms helps in reducing the complexity of algebraic expressions and provides a simpler format that is easier for further mathematical manipulation.

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